Restored one of the assignments from Moodle 2.7.3 (Build: 20141110) into an earlier Moodle 2.7+ (Build: 20140626) , and in there everything works.
I'm about to upgrade that Moodle 2.7.3 (Build: 20141110) to the latest Moodle 2.7.4+ , and see if the problem disappears.
I can't think that any of these settings on my Moodle 2.7.3 would be to blame? :
Assignment default settings (the only non-defaults I set):
Messaging (not sure if this is even remotely connected):
The Moodle 2.7.3+ is the only version of Moodle so far where I have seen this apparent glitch.